Topos theoretic aspects of semigroup actions

Jonathon Funk and Pieter Hofstra

We define the notion of a torsor for an inverse semigroup, which is based
on semigroup actions, and prove that this is precisely the structure
classified by the topos associated with an inverse semigroup. Unlike in
the group case, not all set-theoretic torsors are isomorphic: we shall
give a complete description of the category of torsors. We explain how a
semigroup prehomomorphism gives rise to an adjunction between a
restrictions-of-scalars functor and a tensor product functor, which we
relate to the theory of covering spaces and E-unitary semigroups. We
also interpret for semigroups the Lawvere-product of a sheaf and
distributio$ and finally, we indicate how the theory might be extended to
general semigroups, by defining a notion of torsor and a classifying topos
for those.